Designing and optimizing spiral inductors on silicon has been a painstaking and costly experience, one associated with consecutive silicon spins and several hours of wait with electromagnetic (EM) simulators. Often, the related effort cannot be ported to new silicon processes; moving to a new process with a different metal stack, minimum feature size and grid, means that any spiral optimized at the previous process node will most probably be off in the new one. Overdesign does not apply here; key spiral inductor properties such as quality factor and resonance frequency are largely dependent on parasitics, and these change irregularly between process nodes, especially in the sub-100 nm region.
A variety of methodologies and Electronic Design Automation (EDA) tools have been developed to facilitate synthesis and optimization of inductors in integrated circuits. Each one, however, comes with certain disadvantages.
The methodology of geometric programming (GP) uses a lumped-element inductor model, where each of the parameters is represented by a posynomial expression. The optimization problem with objective (inductance at operating frequency) and constraints (quality factor, self resonance frequency, area) in posynomial form is expressed as a geometric program in convex form that can be solved globally. The major disadvantage of this method is that it requires prior knowledge of the mathematical expressions relating each geometric parameter to inductor performance; this cannot be achieved fully or is impractical in the context of modern silicon processes, where multiple design and manufacturing parameters come into play in the fabrication of an integrated inductor. For instance, there exists no known mathematical formula for relating inductor quality factor to the number and properties of metal layers that can be stacked to construct the inductor's tracks.
Some methodologies in the prior art synthesize and optimize inductors based on ‘trained’ equivalent lumped-element inductor models. Models are trained using either measurements or simulation results that come from full-wave EM solvers or quasi-static EM solvers. The accuracy of the methodology strongly depends on the accuracy of the measurements or simulation results and on the type of the lumped-element model. A major drawback of this approach is that the model should be separately trained for each manufacturing process or process node and is not globally valid for all processes. This leads to increased setup times, poor re-usability between silicon processes, and other inefficiencies.
Other methodologies rely on physical lumped-element inductor models for the synthesis and optimization procedure. The lumped element values are computed from analytic expressions derived from the underlying physics and parameters related to the process, while measured data are only used for validation. For every process several thousands of geometries are modeled and simulated. The geometric parameters and the simulation results are stored in a database. The synthesis and optimization tool retrieves from the database and delivers the optimal geometry that satisfies the objective and constraints of the problem. The method is fast; its accuracy however depends on the accuracy of the model and the size of the database. Also, this method is by definition not re-useable between processes.